Evanescent field trapping and propulsion of Janus particles along optical nanofibers

Small composite objects, known as Janus particles, drive sustained scientific interest primarily targeted at biomedical applications, where such objects act as micro- or nanoscale actuators, carriers, or imaging agents. A major practical challenge is to develop effective methods for the manipulation of Janus particles. The available long-range methods mostly rely on chemical reactions or thermal gradients, therefore having limited precision and strong dependency on the content and properties of the carrier fluid. To tackle these limitations, we propose the manipulation of Janus particles (here, silica microspheres half-coated with gold) by optical forces in the evanescent field of an optical nanofiber. We find that Janus particles exhibit strong transverse localization on the nanofiber and much faster propulsion compared to all-dielectric particles of the same size. These results establish the effectiveness of near-field geometries for optical manipulation of composite particles, where new waveguide-based or plasmonic solutions could be envisaged.


Supplementary Note 1. Optomechanical simulations
shows the theoretical model we used for a spherical silica particle (SP) or Janus particle (JP, represented by a SP half-coated with gold) interacting with the evanescent light field near a single-mode optical nanofiber (ONF). Once the field distributions and the Maxwell's stress tensor, ↔ T , over the computational volume are found (see Methods section in the main text), we perform surface integration (directly in COMSOL) of the elementary force, dF, and torque, dN, presented as where T x = −ewbe.dnTx, T y = −ewbe.dnTy, T z = −ewbe.dnTz are the COMSOL expressions for the x, y, and z components of the "Maxwell downward (i. e. inwardly directed) surface stress tensor", respectively. The "−" sign in the expressions for T x,y,z appears because we follow [1] and define the normal unit vector,n, as outwardly directed with respect to the surface element, dS.
As shown in Supplementary Figure 1b, only the x component of the optical torque on a JP is nonzero, N = N x = xN x = 0 (more precisely, |N x /N y | ≈ |N x /N z | 10 5 ), hence the particle can only spin in the yz plane. In fact, we F  Figure 2. a Total torque on a Janus particle calculated using Eq. S.3 with different values of the friction coefficient, µ. The balance orientation angle, α0, increases with µ. b Balance orientation angle, α0, versus the friction coefficient for different coating thickness, d. Markers (placed at µ = 0.25 corresponding to the best fit between the theory and the experiment) indicate the measured values of α0. Error bars in b are the same as in Fig. 3c in the main text.
get N x < 0 for any orientation angle of a JP, α, and this result agrees with the rotation one would expect from the direction-locked transverse spin angular momentum (SAM) carried by the evanescent field [2,3]. It is worth noting that if the ONF mode broke the mirror symmetry of the system, we should expect to see the impact of the other torque components. In particular, in the case of a circularly polarized input beam, the JP would experience N z = 0 due to the longitudinal SAM of light. In the case of a tilted linear polarization, a vertical "electric" component of SAM would appear [4], forcing the particle to rotate around the y axis under N y = 0. However, in practice, this rotation would be transient and would only occur while the JP is being pulled into the higher-intensity region (that is the polarization plane) of the field, before settling at a certain orientation defined by the optical forces, gravity, and friction. Now let us discuss the dynamic friction force, F fr . It opposes the propulsion of the particle when it is pressed to the ONF by the radial optical force, F y . Since F fr is applied at the particle-nanofiber contact point, it produces a torque, N fr = R p y × F fr , which counteracts the optical torque, N x . The total torque acting on a JP is therefore where µ ∈ [0, 1] is the friction coefficient. Under the action of N Σ , the particle keeps rotating until it reaches a balance orientation at an angle α 0 , which corresponds to N Σ = 0 and dN Σ /dα < 0 (the restorative torque condition). As shown in Supplementary  In addition, we check if the variability of R p (particle radius) and δ (particle-nanofiber separation) could have a significant impact on our results. These parameters could not be measured directly using our experimental approach, hence we performed test simulations with R p = {1.4, 1.5, 1.6} µm (since 2R p = 3.13 ± 0.20 µm, according to the manufacturer) and δ = {10, 100, 150} nm. As one can see in Supplementary Figure 3, theoretical curves for the optical forces and total torques values at variable R p (and fixed δ = 10 nm) or variable δ (and fixed R p = 1.5 µm) are somewhat different; yet, the balance orientation of the gold cap, α 0 , is almost unchanged, and the corresponding values of F z (and the propulsion enhancement, ξ z ) vary by less than 2%. This uncertainty is about ten times smaller than the standard deviation ranges in our experiments. Therefore, we believe that the main source of error was the nanoscale variability of the gold coating.   Radial and longitudinal optical forces (a) and the total torque (b) on a Janus particle with a variable radius, Rp, and a fixed distance to the nanofiber, δ = 10 nm. c, d Same for a variable δ and fixed radius, Rp = 1.5 µm. Insets in c, d: balance orientation of the golden cap (found at NΣ = 0) vs. the particle radius and the particle-nanofiber separation distance, respectively. In all simulations we used R f = 0.35 µm, d = 20 nm, and µ = 0.25.

Supplementary Note 2. Thermal simulations
Once the optical field distribution in the particle-nanofiber system is found, we can simulate the temperature increase resulting from the conversion of electromagnetic energy into heat and its evolution in time. These simulations are much faster compared to those of light scattering, thus we extended the simulation domain to  Figure 4b-d) indicate that light-induced heating around an uncoated particle is negligibly small, whereas ∆T reaches 90 K at an input power of 10 mW (notably, the maximum of ∆T is higher for the thinner coating) and exceeds 500 K at 100 mW (Supplementary Figure 5). Assuming the pressure to remain constant, we should expect a cavitation bubble to be created near the gold cap once ∆T > 80 K. However, no cavitation has been seen in our experiments with JPs near an ONF. Such a strong discrepancy between theoretical and experimental results is quite surprising given that the heat transfer model in COMSOL Multiphysics is the common method applied for analysis of light-induced thermal effects, see for instance [7][8][9] where good agreement was found between simulations and measurements with laser-illuminated gold films and nanostructures in water. One possible reason for the overestimation of ∆T in our case is the fact that the geometry in the thermal model is static, whereas in reality the gold cap on a JP only assumes its stable orientation when the particle is being propelled along the nanofiber. In principle, propulsion can be incorporated into the model by iterative displacement of the thermal distribution. However, we argue that such a complication would not be necessary because the particle's motion is much slower compared to the stabilization of the heat distribution. Indeed, as one can see in Supplementary  Figure 4f, g, ∆T saturates in less than 0.5 ms, during which time the particle would move by 6.2 nm (according to the linear fit of the experimental points in Fig. 4d of the main text). This displacement corresponds to only 0.2% of the particle's radius, therefore we can safely neglect the JP's propulsion in the thermal model. Notably, the saturation time is virtually independent of the optical power, whereas ∆T at saturation is proportional to the power, as shown in Supplementary Figure 5d.
In order to find a more plausible explanation for the unrealistically high values of ∆T in the simulations, we recall that in our optical model the metallic coating is treated as a perfectly smooth layer of crystalline gold, while in practice the conditions for vapor deposition may vary with the angle between the surface normal and the velocity of the gold atoms. Vapor deposition of gold on silica substrates without any special treatment (as in our experiments) is known to produce films with nanoscale imperfections [10], which can only get more prominent as the surface deviates from a plane. Moreover, particles that happened to be tightly packed during the coating have noticeable defects at the edges of the gold cap (see Fig. 1a, b of the main text). Therefore, the actual properties of the material near the edges of the gold cap on a JP may significantly differ from those assumed in the numerical simulations. In particular, obstructions to the propagation of surface plasmons in these areas would lead to an increase of scattering and the reduction of absorption of light, hence the actual light-induced heating might be much smaller than that calculated for a perfect golden film. Due to the size limitations of our 3D model, we cannot include nanoscale imperfections of the coating. Instead, we tried to mimic the lack of absorption by simply removing gold from the areas located further than R cap ≤ R p from the symmetry axis of the particle. As shown in Fig.7 of the main text, a reduction of the effective radius of the gold cap,R = R cap /R p , by a few percent would result in a dramatic decrease of ∆T (as well as the optical forces, F z and F y , and the propulsion enhancement, ξ z ) to values below the boiling temperature of water. Importantly, the reduction ofR would also affect the spatial distribution of ∆T (compare Fig. 7a and Fig. 7b in the main text) and thus the temperature gradient, ∇T . Without the knowledge of these distributions we cannot accurately calculate thermal forces and torques on a JP near an ONF. One should also note that such calculations rely on surface integration of the fluid slip velocity, v = D T ∇ T , where ∇ T is the temperature gradient tangential to the surface, and D T is the thermodiffusion coefficient which is a function of multiple parameters of the particle-fluid system [11]. Evaluation of D T for our experimental conditions is a very challenging task that is beyond the scope of the present study.   Figure 7. Measurements of the gold cap orientation angle. The gold cap orientation can be directly seen when the JP is propelled under horizontal (E y) polarization of the beam coupled to the ONF. To measure the orientation angle, α0, we extract a random video frame from the middle of the recorded track (in order to have the particle illuminated symmetrically). The frame is imported into Inkscape (an open-source graphics software) where we increase the contrast (by 20% for d = 10 nm and 40% for d = 20 nm) and lightness (0-10% and 10-20%, respectively) of the image, and then determine the angle using the Measurement Tool. In order to improve the accuracy of these measurements, we repeated them for 3 to 5 different JPs per experimental sample. We also tried propelling some particles by beam 2 (towards z < 0) and found that the gold cap orientation with respect to the propagation direction was preserved, thereby confirming that the illumination was indeed uniform. The above images correspond to the experimental data in Fig. 5c of the main text.